Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, K

Average Error: 21.1 → 18.4

Time: 15.9s

Precision: 64

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r688158 = 2.0;
        double r688159 = x;
        double r688160 = sqrt(r688159);
        double r688161 = r688158 * r688160;
        double r688162 = y;
        double r688163 = z;
        double r688164 = t;
        double r688165 = r688163 * r688164;
        double r688166 = 3.0;
        double r688167 = r688165 / r688166;
        double r688168 = r688162 - r688167;
        double r688169 = cos(r688168);
        double r688170 = r688161 * r688169;
        double r688171 = a;
        double r688172 = b;
        double r688173 = r688172 * r688166;
        double r688174 = r688171 / r688173;
        double r688175 = r688170 - r688174;
        return r688175;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r688176 = y;
        double r688177 = z;
        double r688178 = t;
        double r688179 = r688177 * r688178;
        double r688180 = 3.0;
        double r688181 = r688179 / r688180;
        double r688182 = r688176 - r688181;
        double r688183 = cos(r688182);
        double r688184 = 0.9999999999999932;
        bool r688185 = r688183 <= r688184;
        double r688186 = 2.0;
        double r688187 = x;
        double r688188 = sqrt(r688187);
        double r688189 = r688186 * r688188;
        double r688190 = r688178 * r688177;
        double r688191 = sqrt(r688180);
        double r688192 = 2.0;
        double r688193 = pow(r688191, r688192);
        double r688194 = r688190 / r688193;
        double r688195 = cos(r688194);
        double r688196 = exp(r688195);
        double r688197 = log(r688196);
        double r688198 = 1.0;
        double r688199 = r688198 * r688176;
        double r688200 = cos(r688199);
        double r688201 = r688197 * r688200;
        double r688202 = sin(r688199);
        double r688203 = -r688194;
        double r688204 = sin(r688203);
        double r688205 = r688202 * r688204;
        double r688206 = r688201 - r688205;
        double r688207 = 3.0;
        double r688208 = pow(r688206, r688207);
        double r688209 = cbrt(r688208);
        double r688210 = r688178 / r688191;
        double r688211 = -r688210;
        double r688212 = r688177 / r688191;
        double r688213 = r688210 * r688212;
        double r688214 = fma(r688211, r688212, r688213);
        double r688215 = cos(r688214);
        double r688216 = r688209 * r688215;
        double r688217 = -r688213;
        double r688218 = fma(r688198, r688176, r688217);
        double r688219 = sin(r688218);
        double r688220 = sin(r688214);
        double r688221 = r688219 * r688220;
        double r688222 = r688216 - r688221;
        double r688223 = r688189 * r688222;
        double r688224 = a;
        double r688225 = b;
        double r688226 = r688225 * r688180;
        double r688227 = r688224 / r688226;
        double r688228 = r688223 - r688227;
        double r688229 = 0.5;
        double r688230 = pow(r688176, r688192);
        double r688231 = r688229 * r688230;
        double r688232 = r688198 - r688231;
        double r688233 = r688189 * r688232;
        double r688234 = r688233 - r688227;
        double r688235 = r688185 ? r688228 : r688234;
        return r688235;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	21.1
Target	19.1
Herbie	18.4

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999932

   1. Initial program 20.3

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]

   4. Applied times-frac 20.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \color{blue}{\frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}}\right) - \frac{a}{b \cdot 3}\]

   5. Applied add-sqr-sqrt 45.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\color{blue}{\sqrt{y} \cdot \sqrt{y}} - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\]

   6. Applied prod-diff 45.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right) + \mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)} - \frac{a}{b \cdot 3}\]

   7. Applied cos-sum 45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right)} - \frac{a}{b \cdot 3}\]

   8. Simplified 42.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)} - \sin \left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   9. Simplified 20.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \color{blue}{\sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)}\right) - \frac{a}{b \cdot 3}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   12. Simplified 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{\color{blue}{{\left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)\right)}^{3}}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   13. Using strategy rm

   14. Applied fma-udef 20.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\cos \color{blue}{\left(1 \cdot y + \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   15. Applied cos-sum 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\color{blue}{\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   16. Simplified 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\color{blue}{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) \cdot \cos \left(1 \cdot y\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   17. Using strategy rm

   18. Applied add-log-exp 19.8

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\color{blue}{\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right)} \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9999999999999932 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.5

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]

   2. Taylor expanded around 0 15.8

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999999999999932276395497865451034158468:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{{\left(\log \left(e^{\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right)\right)}^{3}} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt{3}}, \frac{z}{\sqrt{3}}, \frac{t}{\sqrt{3}} \cdot \frac{z}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))